Let $S=\{1,2,3,4,5,6\}$ and $X$ be the set of all relations $R$ from $S$ to $S$ that satisfy both the following properties:
$i$. $R$ has exactly $6$ elements.
$ii$. For each $(a, b) \in R$, we have $|a-b| \geq 2$.
Let $Y=\{R \in X$ : The range of $R$ has exactly one element $\}$ and $Z=\{R \in X: R$ is a function from $S$ to $S\}$.
Let $n(A)$ denote the number of elements in a Set $A$.
(There are two questions based on $PARAGRAPH " 1 "$, the question given below is one of them)
($1$) If $n(X)={ }^m C_6$, then the value of $m$ is. . . .
($2$) If the value of $n(Y)+n(Z)$ is $k^2$, then $|k|$ is. . . .
Give the answer or quetion ($1$) and ($2$)
Let $S=\{1,2,3,4,5,6\}$ and $X$ be the set of all relations $R$ from $S$ to $S$ that satisfy both the following properties:
$i$. $R$ has exactly $6$ elements.
$ii$. For each $(a, b) \in R$, we have $|a-b| \geq 2$.
Let $Y=\{R \in X$ : The range of $R$ has exactly one element $\}$ and $Z=\{R \in X: R$ is a function from $S$ to $S\}$.
Let $n(A)$ denote the number of elements in a Set $A$.
(There are two questions based on $PARAGRAPH " 1 "$, the question given below is one of them)
($1$) If $n(X)={ }^m C_6$, then the value of $m$ is. . . .
($2$) If the value of $n(Y)+n(Z)$ is $k^2$, then $|k|$ is. . . .
Give the answer or quetion ($1$) and ($2$)
- [IIT 2024]
- A
$20,36$
- B
$20,38$
- C
$20,40$
- D
$20,45$
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